On semilattice-based logics with an algebraizable assertional companion (Q2888156)

The paper studies semilattice-based logics from an abstract algebraic logic point of view. A logic is ''semilattice-based'' if it is the logic associated to the semilattice order of a variety of algebras having a semilattice reduct. If the semilattice order has a top element 1, then another logic can be naturally associated to the above-mentioned variety of algebras, namely, the logic whose set of designated elements is exactly {1}. This is called the ``assertional companion'' of the semilattice-based logic. Examples of such pairs can be found, e.g., in modal logic (local and global consequence relations) and in many-valued logic.NEWLINENEWLINESemilattice-based logics are, by definition, self-extensional. The paper shows that, under the additional assumption that the assertional companion is algebraizable, some nice properties follow. In a certain sense, the semilattice-based logic behaves as if it were protoalgebraic, although it need not be so. Another interesting result is that the two above-mentioned properties (being, respectively, self-extensional and algebraizable) are exactly the ones that separate the two logics, in the sense that, if the semilattice-based logic is algebraizable or the assertional companion is self-extensional, then the two logics coincide (Theorem 15).